Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model

A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remain...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2022
Main Authors:	Marquette, Ian, Quesne, Christiane
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Language:	English
Published:	Інститут математики НАН України 2022
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Cite this:	Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model. Ian Marquette and Christiane Quesne. SIGMA 18 (2022), 005, 24 pages

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author	Marquette, Ian Quesne, Christiane
author_facet	Marquette, Ian Quesne, Christiane
citation_txt	Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model. Ian Marquette and Christiane Quesne. SIGMA 18 (2022), 005, 24 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remain to be studied. We present a set of six operators { ⁺⁻, ⁺⁻, ⁺⁻} that can be combined to build a (3) hidden algebra. The latter can be embedded in an (6) algebra, as well as in an (1/6) superalgebra. The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state. We present the action of these operators and study the construction of an extended biorthogonal basis. These rely on establishing various nontrivial polynomial and commutator identities. We also make a connection between the hidden symmetry and the underlying superintegrability property of the model. Interestingly, the integrals generate a cubic algebra. This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 005, 24 pages Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Models with Quadratic Complex Interaction. II. Three-Dimensional Model Ian MARQUETTE a and Christiane QUESNE b a) School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia E-mail: i.marquette@uq.edu.au b) Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium E-mail: christiane.quesne@ulb.be Received September 01, 2021, in final form January 03, 2022; Published online January 14, 2022 https://doi.org/10.3842/SIGMA.2022.005 Abstract. A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remained to be studied. We present a set of six operators {A±, B±, C±} that can be combined to build a gl(3) hidden algebra. The latter can be embedded in an sp(6) algebra, as well as in an osp(1/6) superalgebra. The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state. We present the action of these operators and study the construction of an extended biorthogonal basis. These rely on establishing various nontrivial polynomial and commutator identities. We also make a connection between the hidden symmetry and the underlying superintegrability property of the model. Interestingly, the integrals generate a cubic algebra. This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences. Key words: quantum mechanics; complex potentials; pseudo-Hermiticity; Lie algebras; Lie superalgebras 2020 Mathematics Subject Classification: 81Q05; 81Q60; 81R12; 81R15 1 Introduction Over the years, a large body of literature has been devoted to non-Hermitian Hamiltonians with a real spectrum, in particular PT-symmetric systems [3, 4, 5] or, more generally, pseudo- Hermitian ones, for which ηHη−1 = H† with η a Hermitian invertible operator [18, 20]. The concept of pseudo-Hermiticity was introduced a long time ago by Pauli as generalized Hermitic- ity [22] and later on by Scholtz, Geyer, and Hahne as quasi-Hermiticity [23]. However, mainly the one-dimensional case has been explored and only a few two- and three- dimensional systems have been obtained (see, e.g., [2, 6, 8, 13, 21]) and their study may still be incomplete. In the previous paper of this series [15], we re-examined a non-Hermitian two-dimensional system [8] which is exactly solvable due to its shape invariance [1, 7, 10, 12, 14], although mailto:i.marquette@uq.edu.au mailto:christiane.quesne@ulb.be https://doi.org/10.3842/SIGMA.2022.005 2 I. Marquette and C. Quesne it is neither separable nor diagonalizable. It was demonstrated that this model has a gl(2) hidden symmetry algebra and that an underlying sp(4) algebra [16] can be constructed, as well as an osp(1/4) superalgebra [11]. In contrast with the two-dimensional Hermitian harmonic oscillator, however, the algebraic structure is related to integrability, but not superintegrability. The main purpose of the present paper is to consider another non-Hermitian system with a quadratic complex interaction, which is exactly solvable, although not separable nor diagonal- izable [2]. This system being in three dimensions has a more complicated structure of the Jordan blocks that are needed to form a complete basis. This is the reason it deserves a separate study. Another purpose of this paper is to highlight properties of non-Hermitian systems in higher dimensions, such as exact solvability without separation of variables and hidden symmetry that does not lead to superintegrability in the usual way. We would also like to emphasize that the ladder operators constructed from partial differential operators may have distinct properties, such as the possibility of infinitely zero modes. The paper is organized as follows. In Section 2, we review the Hamiltonian of the three- dimensional nonseparable oscillator of [2], as well as its spectrum and wavefunctions Ψk,n,0(x), and we present the set of known ladder operators A±, Q±, which do not close in a finite- dimensional Lie algebra. In Section 3, we construct two additional sets of ladder operators, B± and C±, motivated by their action on the wavefunctions with k = 0. In Section 4, we use them to build a set of nine bilinear operators satisfying a nine-dimensional Lie algebra, which can be transformed into gl(3) and proved to provide the Hamiltonian hidden symmetry algebra. A set of bosonic operators in a nonstandard realization is constructed and used to embed gl(3) in an sp(6) algebra and an osp(1/6) superalgebra. The integrals of motion of [2] are also interpreted in this context to relate the superintegrability of the model to the hidden symmetry algebra. In Section 5, we demonstrate how we can induce the Ψk,n,m(x) associated functions in different algebraic ways. We also present the action of the ladder operators and of the gl(3) linear Casimir operator on all the states belonging to Jordan blocks. The construction of an extended biorthogonal basis [17, 19] is then discussed in Section 6. Finally, Section 7 contains the conclusion. 2 Shape invariant model with quadratic complex interaction Let us consider the three-dimensional model with complex oscillator Hamiltonian [2] H = −∂21 − ∂22 − ∂23 + λ2 ( x21 + x22 + x23 ) + g2 ( x21 − 2ix1x2 − x22 ) − 4λg(x1 − ix2)x3 − 3λ, where g and λ are two real parameters such that λ > \|g\|. This Hamiltonian can be rewritten as H = −4∂z∂z̄ − ∂23 + λ2 ( zz̄ + x23 ) + g2z̄2 − 4λgz̄x3 − 3λ, (2.1) where z = x1 + ix2 and z̄ = x1 − ix2. It satisfies pseudo-Hermiticity with η chosen as P2, which is the operator changing x2 into −x2. With the operators A± = 2∂z ∓ λz̄. (2.2) the Hamiltonian (2.1) satisfies the intertwining properties HA+ = A+(H + 2λ), A−H = (H + 2λ)A−, showing its self-isospectral shape invariance. These equations can also be expressed in the form of ladder-type relations [H,A±] = ±2λA±. (2.3) Ladder Operators and Hidden Algebras for Shape Invariant. II 3 However, unlike ladder operators for the Hermitian harmonic oscillator, the operators (2.2) satisfy the relation [A−, A+] = 0. (2.4) This algebra can be seen as a Euclidean e(2), which differs from the Heisenberg algebra encoun- tered in the Hermitian case. Another set of ladder operators can be introduced [2], Q± = 4∂z∂z̄ − ∂23 ∓ 2λ(z∂z + z̄∂z̄ − x3∂3)± 4gx3∂z ∓ 2gz̄∂3 + λ2 ( zz̄ − x23 ) − g2z̄2 ∓ λ. (2.5) They satisfy the following commutation relations with H, [H,Q±] = ±4λQ±, (2.6) which can also be interpreted as intertwining relations, characteristic of shape invariance. Ad- ditional commutation relations read [A±, Q∓] = ±4λA±, (2.7) [A±, Q±] = 0, (2.8) [Q−, Q+] = −2R̃1 = 8 ( λH − 2gR1 + 3λ2 ) , (2.9) where R1 = 2∂z∂3 + λz̄(gz̄ − λx3). The Schrödinger equation HΨ(x) = EΨ(x) can be solved in terms of the operators A± and Q± [2]. The (unnormalized) ground-state wavefunction Ψ0, such that A−Ψ0 = Q−Ψ0 = 0, is given by Ψ0(z, z̄, x3) = e− λ 2 (zz̄+x2 3)+gz̄x3 (2.10) with corresponding energy E0 = 0. The operators A+ and Q+ allow to obtain from it the excited-state wavefunctions Ψk,n,0(z, z̄, x3) = ck,n(Q +)k(A+)nΨ0(z, z̄, x3), (2.11) corresponding to Ek,n = 2λ(2k + n). Here, k and n run over 0, 1, 2, . . . , and ck,n is some normalization coefficient. The determination of the latter, which is far more complicated than in the two-dimensional case, will be studied in detail in Section 6. 4 I. Marquette and C. Quesne The action of A± and Q± on the wavefunctions can be easily calculated and is given by A+Ψk,n,0 = ck,n ck,n+1 Ψk,n+1,0, (2.12) A−Ψk,n,0 = − 4λkck,n ck−1,n+1 Ψk−1,n+1,0, (2.13) Q+Ψk,n,0 = ck,n ck+1,n Ψk+1,n,0, Q−Ψk,n,0 = 8λ2k(2n+ 2k + 1)ck,n ck−1,n Ψk−1,n,0 − 16g2k(k − 1)ck,n ck−2,n+2 Ψk−2,n+2,0. The algebra generated by H, A±, and Q±, and whose commutation relations are given in equations (2.3), (2.4), (2.6), (2.7), (2.8), and (2.9), is not a finite-dimensional Lie algebra. If the operators A± and Q± are useful for building the Hamiltonian wavefunctions, they do not lead to the hidden symmetry algebra. Furthermore, the wavefunctions Ψk,n,0(z, z̄, x3) with n ̸= 0 are self-orthogonal, which signals that H is nondiagonalizable, so that some associated functions must be introduced to complete the basis and to get a resolution of identity. To try to solve these problems, it will prove convenient to introduce some additional ladder operators. This will be the purpose of Section 3. 3 Construction of additional sets of ladder operators If we consider the subset {Ψ0,n,0 \|n = 0, 1, 2, . . . } of wavefunctions, it is clear from equa- tions (2.12) and (2.13) that we have a raising operator A+, but no lowering one since A− annihilates Ψ0,n,0. This is actually another difference with the Hermitian case. The fact that the ladder operators are given by differential operators depending on more than one variable allows the existence of infinitely many zero modes. Such a lowering operator is provided by B− = ∂z̄ + λ 2 z − gx3, for which B−Ψ0,n,0 ∝ z̄n−1e− λ 2 (zz̄+x2 3)+gz̄x3 ∝ Ψ0,n−1,0. This allows to consider another operators B+, defined by B+ = ∂z̄ − λ 2 z + gx3. With H, the pair of operators B± satisfy the commutation relations [H,B±] = ±2λB± ∓ 2gC±, (3.1) where there appear two new operators C± = ∂3 ± gz̄ ∓ λx3. The other commutation relations are given by [H,C±] = ∓2gA± ± 2λC±, (3.2) [A−, B+] = [B−, A+] = [C−, C+] = −2λ, (3.3) [B−, C+] = [C−, B+] = 2g. (3.4) Ladder Operators and Hidden Algebras for Shape Invariant. II 5 From the three sets of operators A±, B±, and C±, we can generate the operators Q±, defined in (2.5), since Q± = 2A±B± − (C±)2. (3.5) From equation (3.5) and the commutation relations (2.4), (3.3), and (3.4), it follows that [B±, Q±] = [C±, Q±] = 0, [B±, Q∓] = ±4λB∓ ± 4gC∓, [C±, Q∓] = ∓4gA∓ ∓ 4λC∓, together with the set of relations (2.7), (2.8) and (2.9). In contrast with the algebra generated by H, A±, and Q±, the one generated by H, A±, B±, C±, and whose commutation relations are given by equations (2.3), (2.4), (3.1), (3.2), (3.3), and (3.4), is a finite-dimensional Lie algebra. It has, however, a rather complicated structure, so that some additional transformations have to be carried out, as shown in Section 4. 4 Construction of the hidden symmetry algebra In order to get more insight into the structure of the hidden symmetry algebra of this non- Hermitian Hamiltonian, let us introduce a set of nine bilinear operators R = A+A−, S = B+B−, T = C+C−, U = A+B− +B+A−, V = A+C− + C+A−, W = B+C− + C+B−, (4.1) X = A+B− −B+A−, Y = A+C− − C+A−, Z = B+C− − C+B−, which have the following differential operator realizations R = 4∂2z − λ2z̄2, S = ∂2z̄ − λ2 4 z2 + λgzx3 − g2x23, T = ∂23 − g2z̄2 + 2λgz̄x3 − λ2x23 + λ, U = 4∂z∂z̄ − λ2zz̄ + 2λgz̄x3 + 2λ, V = 4∂z∂3 + 2λgz̄2 − 2λ2z̄x3, W = 2∂z̄∂3 + λgzz̄ − λ2zx3 − 2g2z̄x3 + 2λgx23 − 2g, X = 2(λz − 2gx3)∂z − 2λz̄∂z̄, Y = −4(gz̄ − λx3)∂z − 2λz̄∂3, Z = −2(gz̄ − λx3)∂z̄ − (λz − 2gx3)∂z. They satisfy the commutation relations [R,S] = −2λX, [R, T ] = 0, [R,U ] = 0, [R, V ] = 0, [R,W ] = −2λY, [R,X] = 4λR, [R, Y ] = 0, [R,Z] = −2λV, [S, T ] = 2gZ, [S,U ] = 0, [S, V ] = −2λZ − 2gX, [S,W ] = 0, [S,X] = −4λS, [S, Y ] = −2λW − 2gU, [S,Z] = −4gS, [T,U ] = −2gY, [T, V ] = 2λY, [T,W ] = 2λZ, [T,X] = −2gV, [T, Y ] = 2λV, [T,Z] = 2λW + 4gT, [U, V ] = −2λY, (4.2) [U,W ] = −2λZ + 2gX, [U,X] = 0, [U, Y ] = −2λV − 4gR, [U,Z] = −2λW − 2gU, [V,W ] = −2λX + 2gY, [V,X] = 2λV − 4gR, [V, Y ] = 4λR, [V,Z] = 2λ(U − 2T ) + 2gV, [W,X] = −2λW − 2gU, [W,Y ] = 2λ(U − 2T )− 2gV, [W,Z] = 4λS, [X,Y ] = −2λY, [X,Z] = 2λZ − 2gX, [Y,Z] = 2λX + 2gY, and therefore generate a nine-dimensional Lie algebra. 6 I. Marquette and C. Quesne These bilinear operators can be related to the Hamiltonian H through the equation H = −U − T. This relation, which connects the Hamiltonian with the generators of the nine-dimensional Lie algebra, proves that the latter is a hidden symmetry algebra. Note, however, a distinction with respect to the Hermitian three-dimensional oscillator, for which H is expressed in terms of three commuting components in involution, while here the components U and T have a nonvanishing commutator. We may also point out the commutators of H with the nine operators (4.1), [H,R] = 0, [H,S] = 2gZ, [H,T ] = −2gY, [H,U ] = 2gY, [H,V ] = 0, [H,W ] = −2gX, (4.3) [H,X] = 2gV, [H,Y ] = 4gR, [H,Z] = 2g(U − 2T ), which will be useful in further calculations. 4.1 Connection with gl(3) and bosonic operators We can re-express the nine operators (4.1) in terms of gl(3) generators Eij , i, j = 1, 2, 3, satisfying the commutation relations [Eij , Ekl] = δj,kEil − δi,lEkj . We indeed get the following relations E11 = − 1 2λ T + 1 2 , E22 = λ 2g2 S + 1 2λ T + 1 2g W + 1 2 , E33 = − g2 2λ3 R− λ 2g2 S − 1 2λ T − 1 2λ U − g 2λ2 V − 1 2g W + 1 2 , E12 = i ( 1 2λ T + 1 4g W − 1 4g Z ) , E21 = i ( 1 2λ T + 1 4g W + 1 4g Z ) , E13 = − 1 2λ T − g 4λ2 V − 1 4g W + g 4λ2 Y + 1 4g Z, E31 = − 1 2λ T − g 4λ2 V − 1 4g W − g 4λ2 Y − 1 4g Z, E23 = i ( λ 2g2 S + 1 2λ T + 1 4λ U + g 4λ2 V + 1 2g W − 1 4λ X − g 4λ2 Y ) , E32 = i ( λ 2g2 S + 1 2λ T + 1 4λ U + g 4λ2 V + 1 2g W + 1 4λ X + g 4λ2 Y ) . The gl(3) linear Casimir operator corresponds to E11 + E22 + E33 = − 1 2λ ( T + U + g2 λ2 R+ g λ V − 3λ ) = 1 2λ ( H − g2 λ2 R− g λ V + 3λ ) , (4.4) so that it is a linear combination of the three commuting operators H, R, and V , up to some additive constant. Ladder Operators and Hidden Algebras for Shape Invariant. II 7 Rewriting the generators Eij in terms of A±, B±, and C±, as done in Appendix A, allows to reveal some underlying structure in terms of bosonic operators a±i , i = 1, 2, 3, satisfying the well-known commutation relations [a−i , a + j ] = δi,j , [a±i , a ± j ] = 0. On using the transformation a±1 = i√ 2λ C±, a±2 = 1√ 2λ ( C± + λ g B± ) , a±3 = i√ 2λ ( C± + λ g B± + g λ A± ) , (4.5) it is indeed possible to rewrite Eij as Eij = 1 2 {a+i , a − j } = a+i a − j + 1 2 δij . From the inverse transformation of (4.5), A± = −λ g √ 2λ(a±2 + ia±3 ), B± = g λ √ 2λ(a±2 + ia±1 ), C± = −i √ 2λa±1 , we can also express Q± and H in terms of the bosonic operators, Q± = 2λ [ (a±1 ) 2 − 2(a±2 ) 2 − 2ia±1 a ± 2 + 2a±1 a ± 3 − 2ia±2 a ± 3 ] , H = 2λ [ a+1 a − 1 + 2a+2 a − 2 + i(a+1 a − 2 + a+2 a − 1 )− (a+1 a − 3 + a+3 a − 1 ) + i(a+2 a − 3 + a+3 a − 2 ) ] . The gl(3) hidden symmetry algebra can be embedded into an sp(6) algebra by considering the additional generators [16] D+ ij = 1 2 {a+i , a + j }, D− ij = 1 2 {a−i , a − j }. Together with the bosonic operators, the operators Eij , D + ij , and D− ij then make rise to an osp(1/6) superalgebra [11] (see [15] for more details). Finally, we may also point out the nonstandard differential operator realization of the bosonic operators, a±1 = i√ 2λ (∂3 ± gz̄ ∓ λx3), a±2 = 1√ 2λ ( λ g ∂z̄ + ∂3 ∓ λ2 2g z ± gz̄ ) , a±3 = i√ 2λ ( 2 g λ ∂z + λ g ∂z̄ + ∂3 ∓ λ2 2g z ) . This completes the description of the hidden symmetry algebra. As compared with the Her- mitian case, the analysis carried out here has shown that the Hamiltonian connects in a different way with the algebra Casimir operator and that a nonstandard realization of bosonic operators makes its appearance. Further progress on these ideas might provide a way to classify some classes of nonseparable and nondiagonalizable models. 8 I. Marquette and C. Quesne 4.2 Superintegrability and cubic algebra In [2], four independent operators R0, R1, R2, and R3 commuting with H were identified, the first two being mutually commuting. Here we plan to relate such a superintegrability property with the hidden symmetry algebra. On expressing the operators of [2] in terms of the bilinear operators (4.1), we get R0 = A+A− = R, R1 = 1 8g { 1 2 [Q+, Q−] + 4λH + 12λ2 } = 1 2 V, R2 = 1 8λ [A+A−, Q+Q−] = −R(X − 2λ) + 1 4 {V, Y }, R3 = Q+(A−)2 = R(U −X + 4λ)− 1 4 ( V 2 + Y 2 − {V, Y } ) . From the commutation relations (4.3), it follows that [H,R0] = [H,R1] = [H,R2] = [H,R3] = 0, (4.6) while equation (4.2) leads to [R0, R1] = 0, [R0, R2] = −4λR2 0, [R1, R2] = 2gR2 0, (4.7) and [R0, R3]− 4λR2 0, [R1, R3] = 2gR2 0, [R2, R3] = 8gR1R 2 0 + 4λ(R3 −R2)R0 + 8λR2 1R0. (4.8) Equations (4.6) and(4.7) agree with some results derived in [2], while equation (4.8) is new and shows that the integrals of motion generate a cubic algebra. Note that the superintegrability property of the present model contrasts with what was obtained in [15] for the two-dimensional pseudo-Hermitian oscillator, which was proved to be only integrable. This points out that in the context of non-Hermitian Hamiltonians, the number of integrals of motion may be affected by the structure of the problem. 5 Nondiagonalizability and construction of associated functions As pointed out above, the Hamiltonian H being nondiagonalizable, the excited wavefunc- tions Ψk,n,0 with n ̸= 0 have to be accompanied with some associated functions Ψk,n,m, m = 1, 2, . . . , pn − 1, completing the Jordan blocks. By definition, these functions obey the relation (H − Ek,n)Ψk,n,m = Ψk,n,m−1, m = 1, 2, . . . , pn − 1. (5.1) It is the purpose of the present section to determine them and to establish that the dimension of the Jordan blocks is pn = 2n + 1. We plan to rely on the new sets of ladder operators of Section 3 to provide an algebraic construction of these associated functions. Furthermore, we will also build a subset of them in terms of multivariate polynomials. Finally, we will determine the action of the ladder operators and of the gl(3) Casimir operator on the associated functions. Ladder Operators and Hidden Algebras for Shape Invariant. II 9 5.1 Algebraic construction of associated functions Let us start by noting some equivalences among polynomials of the operators acting on the ground state, (H − 2λ)B+Ψ0 = −2gC+Ψ0, (H − 2λ)2B+Ψ0 = 4g2A+Ψ0 ∝ Ψ0,1,0. Hence (H − 2λ)B+Ψ0 ∝ Ψ0,1,1 and B+Ψ0 ∝ Ψ0,1,2. We would like to extend these results by showing that (H − 2λn)2n(B+)nΨ0 ∝ (A+)nΨ0 ∝ Ψ0,n,0, and more generally that (H − 2λn)2p(B+)nΨ0 = (B+)n−2p p∑ q=0 a(n,p)q (A+B+)q(C+)2p−2qΨ0, p = 0, 1, . . . , n, (5.2) (H − 2λn)2p+1(B+)nΨ0 = (B+)n−2p−1 p∑ q=0 b(n,p)q (A+B+)q(C+)2p+1−2qΨ0, p = 0, 1, . . . , n− 1, (5.3) for some coefficients a (n,p) q and b (n,p) q to be determined. Let us first point out some auxiliary results, (H − 2λn)(A+)p = (A+)p[H − 2λ(n− p)], (H − 2λn)(B+)p = (B+)p[H − 2λ(n− p)]− 2pq(B+)p−1C+, (H − 2λn)(C+)p = (C+)p[H − 2λ(n− p)]− 2pgA+(C+)p−1, which directly follow from the commutation relations established in Sections 2 and 3. By acting on (5.2) with (H − 2λn) and identifying the result with equation (5.3), we get the coefficients b (n,p) q in terms of a (n,p) r , b (n,p) 0 = −2g(n− 2p)a (n,p) 0 , b(n,p)q = −2g [ (2p− 2q + 2)a (n,p) q−1 + (n− 2p+ q)a(n,p)q ] , q = 1, 2, . . . , p. Alternatively, by acting on (5.3) with (H − 2λn) and identifying the result with equation (5.2), where p is replaced by p+ 1, we find the coefficients a (n,p+1) q in terms of b (n,p) r , a (n,p+1) 0 = −2g(n− 2p− 1)b (n,p) 0 , a(n,p+1) q = −2g [ (2p+ 3− 2q)b (n,p) q−1 + (n− 2p+ q − 1)b(n,p)q ] , q = 1, 2, . . . , p, a (n,p+1) p+1 = −2gb(n,p)p . On eliminating b (n,p) q or a (n,p) q between the two sets of relations, we obtain recursion relations for a (n,p) q or b (n,p) q , a (n,p+1) 0 = 4g2(n− 2p− 1)(n− 2p)a (n,p) 0 , a (n,p+1) 1 = 4g2(n− 2p) [ (4p+ 1)a (n,p) 0 + (n− 2p+ 1)a (n,p) 1 ] , a(n,p+1) q = 4g2 [ (2p− 2q + 3)(2p− 2q + 4)a (n,p) q−2 + (n− 2p+ q − 1)(4p− 4q + 5)a (n,p) q−1 + (n− 2p+ q − 1)(n− 2p+ q)a(n,p)q ] , q = 2, 3, . . . , p, a (n,p+1) p+1 = 4g2 [ 2a (n,p) p−1 + (n− p)a(n,p)p ] , 10 I. Marquette and C. Quesne and b (n,p+1) 0 = 4g2(n− 2p− 2)(n− 2p− 1)b (n,p) 0 , b (n,p+1) 1 = 4g2(n− 2p− 1) [ (4p+ 3)b (n,p) 0 + (n− 2p)b (n,p) 1 ] , b(n,p+1) q = 4g2 [ (2p− 2q + 4)(2p− 2q + 5)b (n,p) q−2 + (n− 2p+ q − 2)(4p− 4q + 7)b (n,p) q−1 + (n− 2p+ q − 2)(n− 2p+ q − 1)b(n,p)q ] , q = 2, 3, . . . , p, b (n,p+1) p+1 = 12g2 [ 2b (n,p) p−1 + (n− p− 1)b(n,p)p ] , respectively. The solutions of the recursion relations are given by a(n,p)q = (2g)2p n! (n− 2p+ q)! p! q!(p− q)! (2p− 1)!! (2p− 2q − 1)!! (5.4) and b(n,p)q = (−2g)2p+1 n! (n− 2p+ q − 1)! p! q!(p− q)! (2p+ 1)!! (2p− 2q + 1)!! . (5.5) At this stage, it is worth noting that in equations (5.2) and (5.3), the summations over q do not really go from 0 to p because the exponents of the operators have to be nonnegative, hence q may not be smaller than 2p− n and 2p+ 1− n, respectively. In fact, the summations run from max(0, 2p−n) to p or max(0, 2p+1−n) to p. This property is accounted for by the presence of (n− 2p+ q)! or (n− 2p+ q− 1)! in the denominator for a (n,p) q or b (n,p) q . We may indeed interpret these factorials as Γ(n− 2p+ q + 1) or Γ(n− 2p+ q), which become infinite for q ≤ 2p− n− 1 or q ≤ 2p− n and produce the vanishing of the corresponding a (n,p) q or b (n,p) q . More generally, on applying (Q+)k to both sides of equations (5.2) and (5.3) and taking into account that [H,Q+] = 4λQ+ and [Q+, A+] = [Q+, B+] = [Q+, C+] = 0, we obtain [H − 2λ(n+ 2k)]2p(B+)n(Q+)kΨ0 = (B+)n−2p p∑ q=0 a(n,p)q (A+B+)q(C+)2p−2q(Q+)kΨ0, p = 0, 1, . . . , n, and [H − 2λ(n+ 2k)]2p+1(B+)n(Q+)kΨ0 = (B+)n−2p−1 p∑ q=0 b(n,p)q (A+B+)q(C+)2p+1−2q(Q+)kΨ0, p = 0, 1, . . . , n− 1, so that in particular [H − 2λ(n+ 2k)]2n(B+)nQ+)kΨ0 = a(n,n)n (A+)n(Q+)kΨ0 ∝ Ψk,n,0. (5.6) with a(n,n)n = (2g)2nn!(2n− 1)!!. This establishes that the dimension of the Jordan blocks is pn = 2n+1 and that the associated functions can be obtained from the ground state in the form Ψk,n,m = dk,n[H − 2λ(n+ 2k))]2n−m(B+)n(Q+)kΨ0, m = 1, 2, . . . , 2n. (5.7) Ladder Operators and Hidden Algebras for Shape Invariant. II 11 Their detailed expressions in terms of the ladder operators are given by Ψk,n,m =  dk,n(B +)2µ−n n−µ∑ q=max(0,n−2µ) a(n,n−µ) q (A+B+)q × (C+)2n−2µ−2q(Q+)kΨ0 if m = 2µ, dk,n(B +)2µ+1−n n−µ−1∑ q=max(0,n−2µ−1) b(n,n−µ−1) q (A+B+)q × (C+)2n−2µ−2q−1(Q+)kΨ0 if m = 2µ+ 1, (5.8) where the coefficients are expressed in (5.4) and (5.5), and p has been replaced by n − µ or n−µ−1, respectively. The additional factor dk,n is a normalization coefficient, whose calculation will be discussed in Section 6. The construction of the associated functions Ψk,n,m is displayed in Figure 1. At this stage, it is worth observing that, in accordance with equations (5.1) and (5.6), the wavefunctions (2.11) can be alternatively expressed in the form (5.7) or (5.8), with m set equal to zero. It follows that the normalization coefficients ck,n and dk,n of equations (2.11) and (5.7), respectively, are connected by the relation ck,n = dk,na (n,n) n = dk,n(2g) 2nn!(2n− 1)!!. (5.9) 5.2 Some associated functions in terms of multivariate polynomials After presenting an algebraic description of the associated functions, we will now use an alterna- tive approach wherein some of them will be written in terms of polynomials in the variables z, z̄, and x3. To start with, it is useful to introduce a new set of variables, u = z̄, v = −λz + 2gx3, w = gz̄ − λx3, which are directly related to the action of the raising operators A+, B+, and C+ on Ψ0, A+Ψ0 = −2λuΨ0, B+Ψ0 = vΨ0, C+Ψ0 = 2wΨ0. The Jacobian of the transformation is ∂(u, v, w) ∂(z, z̄, x3) = ∣∣∣∣∣∣ 0 1 0 −λ 0 2g 0 g −λ ∣∣∣∣∣∣ = −λ2 and the inverse transformation writes z = 1 λ2 (2g2u− λv − 2gw), z̄ = u, x3 = 1 λ (gu− w). Furthermore, we may introduce a new operator Dp, defined by (H − 2λp)Ψ0 = Ψ0Dp, and whose explicit expression in terms of the new variables is Dp = 4λ∂2uv − 4g2∂2v + 8λg∂2vw − λ2∂2w + 2λ(u∂u + v∂v + w∂w − p)− 4gw∂v + 2λgu∂w. 12 I. Marquette and C. Quesne Ψ000 Ψ010 Ψ011 Ψ012 Ψ020 Ψ021 Ψ022 Ψ023 Ψ024 Ψ100 Ψ110 Ψ111 Ψ112 Ψ120 Ψ121 Ψ122 Ψ123 Ψ124 . . . . . . . . . A+ B+ (H − 2λ) (H − 2λ) A+ B+ (H − 4λ) (H − 4λ) (H − 4λ) (H − 4λ) Q+ A+ B+ (H − 2λ) (H − 2λ) A+ B+ (H − 4λ) (H − 4λ) (H − 4λ) (H − 4λ) Q+ A+ A+ Figure 1. Construction of associated states by using ladder operators. Its action on the monomials ui, vi, and wi is given by Dpu i = uiDp + 2λiui−1(2∂v + u), Dpv i = viDp + ivi−1 ( 4λ∂u − 8g2∂v + 8λg∂w + 2λv − 4gw ) − 4i(i− 1)g2vi−2, Dpw i = wiDp + 2λiwi−1(4g∂v − λ∂w + gu+ w)− i(i− 1)λ2wi−2. Let us consider the set of functions belonging to a Jordan block with k = 0 and any n (or, in other words, those belonging to the lower lattice in Figure 1), Ψ0,n,2n−p = d0,n(H − E0n) pvnΨ0, p = 0, 1, . . . , 2n, and introduce the notation v (n) i = Γ(n+ 1) Γ(n− i+ 1) vn−i, i = 0, 1, . . . , n, Ladder Operators and Hidden Algebras for Shape Invariant. II 13 so that v (n) 0 = vn, v (n) i = n(n− 1) · · · (n− i+ 1)vn−i, with i = 1, 2, . . . , n, and Dnv (n) i = −2λiv (n) i − 4gv (n) i+1w − 4g2v (n) i+2. Then we obtain Ψ0,n,2n−p = d0,nΨ0 2p∑ q=[ p+1 2 ] v(n)q f (n,p)q (u,w), where f (n,p)q (u,w) = ∑ r,s α(n,p,q) r,s urws is a polynomial of total degree 2p − q and total parity (−1)q in the cubic variable u and the linear one w. This means that 3r + s ≤ 2p − q and (−1)3r+s = (−1)q. The coefficients α (n,p,q) r,s satisfy the recursion relation α(n,p+1,q) r,s = λ [ 2(r + s− q)α(n,p,q) r,s + 2q(s+ 1)α (n,p,q) r−1,s+1 − λ(s+ 1)(s+ 2)α (n,p,q) r,s+2 ] + 4 [ −gα(n,p,q−1) r,s−1 + 2λg(s+ 1)α (n,p,q−1) r,s+1 + λ(r + 1)α (n,p,q−1) r+1,s ] − 4g2α(n,p,q−2) r,s , which can be solved for low values of p and q. Some examples of polynomials f (n,p) q (u,w) are presented in Appendix B. 5.3 Action of the ladder operators and of the gl(3) Casimir operator on Ψk,n,m The structure of the functions Ψk,n,m is complicated, but using our algebraic description in terms of the ladder operators A±, B±, and C± allows us to establish explicit formulas for the action of these operators. For A+, we get A+ψk,n,m = α (k,n,m) 0 ψk,n+1,m + α (k,n,m) 1 ψk+1,n−1,m−2, where the coefficients α (k,n,m) 0 and α (k,n,m) 1 are given by α (k,n,m) 0 = dk,n dk,n+1 1 4g2(n+ 1)(2n+ 1) , (5.10) α (k,n,m) 1 = { dk,n dk+1,n−1 n 2n+1 if m ≥ 2, 0 if m = 0 or 1, (5.11) in terms of the normalization coefficients, whose calculation will be discussed in Section 6. This result can be demonstrated by direct calculations. Let us provide some details on the proof. 14 I. Marquette and C. Quesne Case 1: m = 2µ. Assuming n ≥ 2µ, we get A+Ψk,n,2µ = dk,n(B +)2µ−n−1 n−µ∑ q=n−2µ a(n,n−µ) q (A+B+)q+1(C+)2n−2µ−2q(Q+)kΨ0 = α (k,n,2µ) 0 dk,n+1(B +)2µ−n−1 n−µ+1∑ q=n−2µ+1 a(n+1,n−µ+1) q (A+B+)q(C+)2n−2µ−2q+2(Q+)kΨ0 + α (k,n,2µ) 1 dk+1,n−1(B +)2µ−n−1 [ 2 n−µ+1∑ q=n−2µ+2 a (n−1,n−µ) q−1 (A+B+)q(C+)2n−2µ−2q+2 − n−µ∑ q=n−2µ+1 a(n−1,n−µ) q (A+B+)q(C+)2n−2µ−2q+2 ] (Q+)kΨ0. By equating both expressions, we obtain the following constraints to be satisfied: dk,na (n,n−µ) n−2µ = α (k,n,2µ) 0 dk,n+1a (n+1,n−µ+1) n−2µ+1 − α (k,n,2µ) 1 dk+1,n−1a (n−1,n−µ) n−2µ+1 if q = n− 2µ+ 1, dk,na (n,n−µ) q−1 = α (k,n,2µ) 0 dk,n+1a (n+1,n−µ+1) q + α (k,n,2µ) 1 dk+1,n−1 ( 2a (n−1,n−µ) q−1 − a(n−1,n−µ) q ) if n− 2µ+ 2 ≤ q ≤ n− µ, dk,na (n,n−µ) n−µ = α (k,n) 0 dk,n+1a (n+1,n−µ+1) n−µ+1 + 2α (k,n,2µ) 1 dk+1,n−1a (n−1,n−µ) n−µ if q = n− µ+ 1. These are fulfilled due to the definitions of α (k,n,m) 0 , α (k,n,m) 1 , and a (n,n−µ) q . Whenever n < 2µ, a similar procedure leads to the proof of equations (5.10) and (5.11). Case 2: m = 2µ+ 1. Assuming n ≥ 2µ+ 1, we get A+Ψk,n,2µ+1 = dk,n(B +)2µ−n n−µ∑ q=n−2µ b (n,n−µ−1) q−1 (A+B+)q(C+)2n−2µ−2q+1(Q+)kΨ0 = α (k,n,2µ+1 0 dk,n+1(B +)2µ−n n−µ∑ q=n−2µ b(n+1,n−µ) q (A+B+)q(C+)2n−2µ−2q+1(Q+)kΨ0 + α (k,n,2µ+1) 1 dk+1,n−1(B +)2µ−n [ 2 n−µ∑ q=n−2µ+1 b (n−1,n−µ−1) q−1 (A+B+)q(C+)2n−2µ−2q+1 − n−µ−1∑ q=n−2µ b(n−1,n−µ−1) q (A+B+)q(C+)2n−2µ−2q+1 ] (Q+)kΨ0, thus leading to the following set of constraints by equating similar terms: dk,nb (n,n−µ−1) n−2µ−1 = α (k,n,2µ+1) 0 dk,n+1b (n+1,n−µ) n−2µ − α (k,n,2µ+1) 1 dk+1,n−1b (n−1,n−µ−1) n−2µ if q = n− 2µ, dk,nb (n,n−µ−1) q−1 = α (k,n,2µ+1) 0 dk,n+1b (n+1,n−µ) q + α (k,n,2µ+1) 1 dk+1,n−1 ( 2b (n−1,n−µ−1) q−1 − b(n−1,n−µ−1) q ) if n− 2µ+ 1 ≤ q ≤ n− µ− 1, dk,nb (n,n−µ−1) n−µ−1 = α (k,n,2µ+1) 0 dk,n+1b (n+1,n−µ) n−µ + 2α (k,,n,2µ+1) 1 dk+1,n−1b (n−1,n−µ−1) n−µ−1 if q = n− µ. Ladder Operators and Hidden Algebras for Shape Invariant. II 15 These can be shown to be satisfied by using the explicit formulas for α (k,n,m) 0 , α (k,n,m) 1 , and b (n,n−µ) q . After considering the case where n < 2µ + 1 in a similar way, the proof of the action of A+ on the states Ψk,n,m forming Jordan blocks is complete. The action of B+ and C+ can be obtained in the same way by considering appropriate subcases for the index and deriving a set of constraints. We will only present the final results: B+Ψk,n,m = β (k,n,m) 0 Ψk,n+1,m+2 + β (k,n,m) 1 Ψk+1,n−1,m, where β (k,n,m) 0 = dk,n dk,n+1 (m+ 1)(m+ 2) 2(n+ 1)(2n+ 1) , β (k,nm) 1 = dk,n dk+1,n−1 2g2n(2n−m− 1)(2n−m) 2n+ 1 , and C+Ψk,n,m = γ (k,n,m) 0 Ψk,n+1,m+1 + γ (k,n,m) 1 Ψk+1,n−1,m−1, where γ (k,n,m) 0 = − dk,n dk,n+1 m+ 1 2g(n+ 1)(2n+ 1) , γ (k,nm) 1 =  dk,n dk+1,n−1 2gn(2n−m) 2n+ 1 if m ≥ 1, 0 if m = 0. The action of the operators A−, B−, and C− is more complicated, because we have to use their commutation relations with A+, B+, C+, and Q+, given in Sections 2 and 3, but it can be straightforwardly established. The results are given by A−Ψk,n,m = ᾱ (k,n,m) 0 Ψk−1,n+1,m + ᾱ (k,n,m) 1 Ψk,n−1,m−2, where ᾱ (k,n,m) 0 = − dk,n dk−1,n+1 λk g2(n+ 1)(2n+ 1) , ᾱ (k,n,m) 1 = − dk,n dk,n−1 2λn(2k + 2n+ 1) 2n+ 1 if m ≥ 2, 0 if m = 0 or 1, B−Ψk,n,m = β̄ (k,n,m) 0 Ψk−1,n+1,m+1 + β̄ (k,n,m) 1 Ψk−1,n+1,m+2 + β̄ (k,n,m) 2 Ψk,n−1,m−1 + β̄ (k,n,m) 3 Ψk,n−1,m, where β̄ (k,n,m) 0 = dk,n dk−1,n+1 2k(m+ 1) (n+ 1)(2n+ 1) , β̄ (k,n,m) 1 = − dk,n dk−1,n+1 2λk(m+ 1)(m+ 2) (n+ 1)(2n+ 1) , β̄ (k,n,m) 2 = − dk,n dk,n−1 4g2n(2n−m)(2k + 2n+ 1) 2n+ 1 if m ≥ 1, 0 if m = 0, β̄ (k,n,m) 3 = − dk,n dk,n−1 4λg2n(2n−m)(2n−m− 1)(2k + 2n+ 1) 2n+ 1 , 16 I. Marquette and C. Quesne and C−Ψk,n,m = γ̄ (k,n,m) 0 Ψk−1,n+1,m + γ̄ (k,n,m) 1 Ψk−1,n+1,m+1 + γ̄ (k,n,m) 2 Ψk,n−1,m−2 + γ̄ (k,n,m) 3 Ψk,n−1,m−1, where γ̄ (k,n,m) 0 = dk,n dk−1,n+1 k g(n+ 1)(2n+ 1) , γ̄ (k,n,m) 1 = − dk,n dk−1,n+1 2λk(m+ 1) g(n+ 1)(2n+ 1) , γ̄ (k,n,m) 2 =  dk,n dk,n−1 2gn(2k + 2n+ 1) 2n+ 1 if m ≥ 2, 0 if m = 0 or 1, γ̄ (k,n,m) 3 =  dk,n dk,n−1 4λg(2k + 2n+ 1)n(2n−m) 2n+ 1 if m ≥ 1, 0 if m = 0. Finally, let us consider the action of the gl(3) linear Casimir operator, which, from (4.4), can be expressed in terms of the operators H, R = A+A−, and V = A+C− + C+A−. From equation (5.1), it follows that HΨk,n,m = 2λ(2k + n)Ψk,n,m +Ψk,n,m−1, while the results obtained above can be combined to yield RΨk,n,m = − dk,n dk−1,n+2 λk 4g4(n+ 1)(n+ 2)(2n+ 1)(2n+ 3) Ψk−1,n+2,m − λ(4k + 2n+ 3) 2g2(2n− 1)(2n+ 3) Ψk,n,m−2 − dk,n dk+1,n−2 2λn(n− 1)(2k + 2n+ 1) (2n− 1)(2n+ 1) Ψk+1,n−2,m−4 and VΨk,n,m = dk,n dk−1,n+2 k 4g3(n+ 1)(n+ 2)(2n+ 1)(2n+ 3) Ψk−1,n+2,m + 4k + 2n+ 3 2g(2n− 1)(2n+ 3) ψk,n,m−2 + λ g Ψk,n,m−1 + dk,n dk+1,n−2 2gn(n− 1)(2k + 2n+ 1) (2n− 1)(2n+ 1) Ψk+1,n−2,m−4. These equations lead to the following action for the linear Casimir operator (E11 + E22 + E33)Ψk,n,m = ( 2k + n+ 3 2 ) Ψk,n,m. We conclude that, in spite of the presence of Jordan blocks, the Casimir operator has the same action on their member states as on the eigenstates of the Hermitian harmonic oscillator. Ladder Operators and Hidden Algebras for Shape Invariant. II 17 6 Construction of an extended biorthogonal basis The Hamiltonian being pseudo-Hermitian with η = P2, one has to use a new scalar product [3, 4, 18, 20] ⟨Ψ\|η\|Φ⟩ = ⟨⟨Ψ\|Φ⟩⟩ = ∫ ΨΦd3x. For the ground state (2.10), for instance, ⟨⟨Ψ0\|Ψ0⟩⟩ = √( π λ )3 . On the other hand, the nondiag- onalizability of H makes it necessary to add some associated functions to the wavefunctions in order to complete the Jordan blocks, thereby resulting in a set of functions Ψk,n,m, k, n = 0, 1, . . . , m = 0, 1, . . . , 2n. With the corresponding functions Ψ̃k,n,m for H†, they make up an extended biorthogonal basis, whose scalar product is given by [17, 19] ⟨⟨Ψk,n,m\|Ψk′,n′,m′⟩⟩ = ⟨Ψ̃k,n,m\|Ψk′,n′,m′⟩ = ∫ Ψk,n,mΨk′,n′,m′ d3x = δk,k′δn,n′δm,2n−m′ . The purpose of this section is twofold: first, to determine the normalization coefficient dk,n of the functions Ψk,n,m from the relation ⟨⟨Ψk,n,m\|Ψk,n,2n−m⟩⟩ = 1 (6.1) and second, to check the orthogonality of the functions Ψk,n,m and Ψk,n,m′ with m′ ̸= 2n −m. In the calculations, we will rely on the commutation relations between A±, B±, C±, Q± and use the following results: ⟨⟨OΨ0\|O′Ψ0⟩⟩ = ⟨Ψ0\|O†ηO′\|Ψ0⟩, (A+)†η = −ηA−, (B+)†η = −ηB−, (C+)†η = −ηC−, (Q+)†η = ηQ−, A−Ψ0 = B−Ψ0 = C−Ψ0 = Q−Ψ0 = 0. 6.1 Calculation of the normalization coefficient We will first discuss the two special cases where k = 0 or n = 0, then consider the general case. 6.1.1 Case k = 0 and any n The simplest calculation of d0,n corresponds to m = 0 in (6.1). In such a case, we know from equations (5.7) and (5.9) that Ψ0,n,0 = d0,n(2g) 2nn!(2n− 1)!!(A+)nΨ0, Ψ0,n,2n = d0,n(B +)nΨ0. On inserting these expressions in (6.1), it is easy to get the result ⟨⟨Ψ0,n,0\|Ψ0,n,2n⟩⟩ = d20,n ( 8g2λ )n (n!)2(2n− 1)!!⟨⟨Ψ0\|Ψ0⟩⟩, from which it follows that d0,n = [ 23n(n!)2(2n− 1)!!λng2n (π λ )3/2 ]−1/2 . (6.2) As a check, we have proved that the conditions ⟨⟨ψ0,n,2µ\|ψ0,n,2n−2µ⟩⟩ = ⟨⟨ψ0,n,2µ+1\|ψ0,n,2n−2µ−1⟩⟩ = 1 (6.3) for any allowed µ value, lead to the same value for d0,n as given in (6.2). 18 I. Marquette and C. Quesne Let us consider, for instance, ⟨⟨ψ0,n,2µ\|ψ0,n,2n−2µ⟩⟩ = 1 for n ≥ 2µ (or n ≤ 2n − 2µ) and provide some key steps in the proof. From Ψ0,n,2µ = d0,n(B +)2µ−n n−µ∑ q=n−2µ a(n,n−µ) q (A+B+)q(C+)2n−2µ−2qΨ0 and Ψ0,n,2n−2µ = d0,n(B +)n−2µ µ∑ r=0 a(n,µ)r (A+B+)r(C+)2µ−2rΨ0, we get ⟨⟨Ψ0,n,2µ\|Ψ0,n,2n−2µ⟩⟩ = d20,n(−1)n n−µ∑ q=n−2µ µ∑ r=0 a(n,n−µ) q a(n,µ)r × ⟨Ψ0\|η(C−)2n−2µ−2q(B−)2µ−n+q(A−)q(A+)r(B+)n−2µ+r(C+)2µ−2r\|Ψ0⟩, where ⟨Ψ0\|η(C−)2n−2µ−2q(B−)2µ−n+q(A+)r(C+)2µ−2r [ (A−)q, (B+)n−2µ+r ] \|Ψ0⟩ =  0 if q > n− 2µ+ r, (−2λ)q (n− 2µ+ r)! (n− 2µ+ r − q)! ⟨Ψ0\|η(C−)2n−2µ−2q(B−)2µ−n+q ×(A+)r(C+)2µ−2r(B+)n−2µ+r−q\|Ψ0⟩ if q ≤ n− 2µ+ r. This yields ⟨⟨Ψ0,n,2µ\|Ψ0,n,2n−2µ⟩⟩ = d20,n(−1)n n−µ∑ q=n−2µ µ∑ r=q−n+2µ a(n,n−µ) q a(n,µ)r (−2λ)q (n− 2µ+ r)! (n− 2µ+ r − q)! × ⟨Ψ0\|η(C−)2n−2µ−2q(B+)n−2µ+r−q(B−)2µ−n+q(A+)r(C+)2µ−2r\|Ψ0⟩. On using (B−)2µ−n+q(A+)r(C+)2µ−2r\|Ψ0⟩ = δr,2µ−n+q 2µ−n+q∑ s=0 22µ−n+q ( 2µ− n+ q s ) (2µ− n+ q)! s! (2n− 2µ− 2q)! (2n− 2µ− 2q − s)! × (−λ)2µ−n+q−sgs(A+)s(C+)2n−2µ−2q−s\|Ψ0⟩ and ⟨Ψ0\|η(C−)2n−2µ−2q(A+)s(C+)2n−2µ−2q−s\|Ψ0⟩ = δs,0(−2λ)2n−2µ−2q(2n− 2µ− 2q)!⟨⟨Ψ0\|Ψ0⟩⟩, we get ⟨⟨Ψ0,n,2µ\|Ψ0,n,2n−2µ⟩⟩ = d20,n n−µ∑ q=n−2µ a(n,n−µ) q a (n,µ) 2µ−n+q(2λ) n × q!(2µ− n+ q)!(2n− 2µ− 2q)!⟨⟨Ψ0\|Ψ0⟩⟩ Ladder Operators and Hidden Algebras for Shape Invariant. II 19 and, with the explicit values of a (n,n−µ) q and a (n,µ) 2µ−n+q, ⟨⟨Ψ0,n,2µ\|Ψ0,n,2n−2µ⟩⟩ = d20,n(2λ) n(2g)2n(n!)2µ!(n− µ)!(2n− 2µ− 1)!!(2µ− 1)!! × n−µ∑ q=n−2µ Γ(12) q!(2µ− n+ q)!(n− µ− q)!Γ(n− µ− q + 1 2) ⟨⟨Ψ0\|Ψ0⟩⟩. It only remains to use some standard binomial identity to transform the right-hand side of the latter equation into that of equation (6.3), which completes the proof for that case. The other cases can be dealt with in a similar way. 6.1.2 Case n = 0 and any k In this case, equation (6.1) becomes ⟨⟨Ψk,0,0\|Ψk,0,0⟩⟩ = 1, where ⟨⟨Ψk,0,0\|Ψk,0,0⟩⟩ = d2k,0⟨Ψ0\|η(Q−)k(Q+)k\|Ψ0⟩ = d2k,0⟨Ψ0\|η(Q−)k−1 [ Q−, (Q+)k ] \|Ψ0⟩. From the identity [Q−, (Q+)k] = 8k(Q+)k−1 [ λH − gV + (2k + 1)λ2 ] − 16k(k − 1)g2(Q+)k−2(A+)2, (6.4) which can be proved by induction over k, and the identities H\|Ψ0⟩ = V \|Ψ0⟩ = 0, it follows that ⟨⟨Ψk,0,0\|Ψk,0,0⟩⟩ = d2k,0⟨Ψ0\|η(Q−)k−1 { a (k) 1,0(Q +)k−1 + a (k) 1,1(Q +)k−2(A+)2 } \|Ψ0⟩, with a (k) 1,0 = 8k(2k+1)λ2 and a (k) 1,1 = −16k(k−1)g2. Since [Q−, (A+)2]\|Ψ0⟩ = −8λA+A−\|Ψ0⟩ = 0, the latter equation can be rewritten as ⟨⟨Ψk,0,0\|Ψk,0,0⟩⟩ = d2k,0⟨Ψ0\|η(Q−)k−2 { a (k) 1,0[Q −, (Q+)k−1] + a (k) 1,1[Q −, (Q+)k−2](A+)2 } \|Ψ0⟩. On using the identity (6.4) again, we get ⟨⟨Ψk,0,0\|Ψk,0,0⟩⟩ = d2k,0⟨Ψ0\|η(Q−)k−2 × { a (k) 2,0(Q +)k−2 + a (k) 2,1(Q +)k−3(A+)2 + a (k) 2,2(Q +)k−4(A+)4 } \|Ψ0⟩ for some coefficients a (k) 2,0, a (k) 2,1, and a (k) 2,2. On pursuing in the same way, we arrive at the relation ⟨⟨Ψk,0,0\|Ψk,0,0⟩⟩ = d2k,0⟨Ψ0\|η(Q−)k−l min(l,k−l)∑ m=0 a (k) l,m(Q+)k−l−m(A+)2m\|Ψ0⟩, where l = 2, 3, . . . , k and a (k) l,m are some coefficients. For l = k we are left with ⟨⟨Ψk,0,0\|Ψk,0,0⟩⟩ = d2k,0a (k) k,0⟨⟨Ψ0\|Ψ0⟩⟩, where a (k) k,0 = a (k) 1,0a (k−1) 1,0 · · · a(1)1,0 = 8kk!(2k + 1)!!λ2k. We have therefore proved that for any k dk,0 = [ 8kk!(2k + 1)!!λ2k (π λ )3/2 ]−1/2 . (6.5) 20 I. Marquette and C. Quesne 6.1.3 Case any k and any n Let us now turn ourselves to the general case and consider m = 0 in equation (6.1). We plan to prove that provided the three following results T1 ≡ ⟨Ψ0\|η(Q−)k−1(A−)n(A+)2(B+)n(Q+)k−2\|Ψ0⟩ = 0, T2 ≡ ⟨Ψ0\|η(Q−)k−1(A−)nA+(B+)n−1(Q+)k−1\|Ψ0⟩ = 0, T3 ≡ ⟨Ψ0\|η(Q−)k−1(A−)n(B+)n−2(Q+)k\|Ψ0⟩ = 0 are true, then ⟨⟨Ψk,n,0\|Ψk,n,2n⟩⟩ = d2k,n8 k+n2k(n!)2(2n− 1)!!(1)k ( 2n+ 3 2 ) k g2nλ2k+n⟨⟨Ψ0\|Ψ0⟩⟩, (6.6) thus leading to the general expression dk,n = [ 8k+nk!(n!)2(2n+ 1)−1(2n+ 2k + 1)!!g2nλ2k+n (π λ )3/2 ]−1/2 . (6.7) for the normalization coefficient. We note that equation (6.7) agrees with equation (6.2) obtained for k = 0 (as well as with equation (6.5) derived for n = 0). Let us show that if equation (6.7) is valid for k−1 and any n, and T1 = T2 = T3 = 0, then it will be valid for k and n. On starting from Ψk,n,0 = dk,n(2g) 2nn!(2n− 1)!!(A+)n(Q+)kΨ0 and Ψk,n,2n = dk,n(B +)n(Q+)kΨ0, we get ⟨⟨Ψk,n,0\|Ψk,n,2n⟩⟩ = d2k,n(2g) 2nn!(2n− 1)!!⟨Ψ0\| [ (A+)n(Q+)k−1 ]† ηQ−(B+)n(Q+)k\|Ψ0⟩. The identity [Q−, (B+)n] = −4nλ(B+)n−1B− − 4ng(B+)n−1C− − 4n(n− 1)g2(B+)n−2, which is easily proved by induction over n, leads to Q−(B+)n(Q+)k\|Ψ0⟩ = [ (B+)nQ− − 4nλ(B+)n−1B− − 4ng(B+)n−1C− − 4n(n− 1)g2(B+)n−2 ] (Q+)k\|Ψ0⟩ or Q−(B+)n(Q+)k\|Ψ0⟩ = [ 8λ2k(2n+ 2k + 1)(B+)n(Q+)k−1 − 16g2k(k − 1)(A+)2(B+)n(Q+)k−2 − 16g2knA+(B+)n−1(Q+)k−1 − 4g2n(n− 1)(B+)n−2(Q+)k ] \|Ψ0⟩, where, in the last step, we used Q−(Q+)k\|Ψ0⟩ = [ 8λ2k(2k + 1)(Q+)k−1 − 16g2k(k − 1)(A+)2(Q+)k−2 ] \|Ψ0⟩, B−(Q+)k\|Ψ0⟩ = [ −4λkB+(Q+)k−1 − 4gkC+(Q+)k−1 ] \|Ψ0⟩, C−(Q+)k\|Ψ0⟩ = [ 4gkA+(Q+)k−1 + 4λkC+(Q+)k−1 ] \|Ψ0⟩, Ladder Operators and Hidden Algebras for Shape Invariant. II 21 also demonstrated by induction over k. This yields ⟨⟨Ψk,n,0\|Ψk,n,2n⟩⟩ = d2k,n(2g) 2nn!(2n− 1)!! { 8λ2k(2n+ 2k + 1)⟨Ψ0\|η(Q−)k−1(A−)n(B+)n(Q+)k−1\|ψ0⟩ − 16g2k(k − 1)T1 − 16g2knT2 − 4g2n(n− 1)T3 } = d2k,n(2g) 2nn!(2n− 1)!!8λ2k(2n+ 2k + 1) ⟨⟨Ψk−1,n,0\|Ψk−1,n,2n⟩⟩ d2k−1,n(2g) 2nn!(2n− 1)!! , which amounts to equation (6.6) under the assumptions made. 6.2 Orthogonality of the basis Let us start with the simplest case with k = 0 and n = 1. The corresponding Jordan block is spanned by the three states Ψ0,1,0 = d0,14g 2A+Ψ0, Ψ0,1,1 = d0,1(−2g)C+Ψ0, Ψ0,1,2 = d0,1B +Ψ0, where use has been made of equations (5.4), (5.5), (5.8), and d0,1 = [ 8λg2 (π λ )3/2 ]−1/2 in accordance with equation (6.2). Direct calculations, as explained at the beginning of this section, lead to the following results ⟨⟨Ψ0,1,0\|Ψ0,1,0⟩⟩ = ⟨⟨Ψ0,1,0\|Ψ0,1,1⟩⟩ = ⟨⟨Ψ0,1,2\|Ψ0,1,2⟩⟩ = 0, ⟨⟨Ψ0,1,1\|Ψ0,1,2⟩⟩ = 1 2λ , which show that the three states do not form an orthogonal basis, as it was expected. It is therefore necessary to orthogonalize it, which amounts to going from Ψ0,1,m to Φ0,1,m, defined by Φ0,1,0 = Ψ0,1,0, Φ0,1,1 = Ψ0,1,1 − 1 2λ Ψ0,1,0, Φ0,1,2 = Ψ0,1,2, and such that ⟨⟨Φ0,1,m\|Φ0,1,m′⟩⟩ = δm,2−m′ for m,m′ = 0, 1, 2. We have studied in the same way some other Jordan blocks and constructed, from the Ψk,n,m functions, new functions Φk,n,m satisfying the condition ⟨⟨Φk,n,m\|Φk′,n′,m′⟩⟩ = δk,k′δn,n′δm,2n−m′ . For k = 0 and n = 2, for instance, Φ0,2,0 = Ψ0,2,0, Φ0,2,1 = Ψ0,2,1 − 1 2λ Ψ0,2,0, Φ0,2,2 = Ψ0,2,2 + 1 6λ2 Ψ0,2,0 − 1 2λ Ψ0,2,1, Φ0,2,3 = Ψ0,2,3 + 1 48λ3 Ψ0,2,0 − 1 24λ2 Ψ0,2,1, Φ0,2,4 = Ψ0,2,4, 22 I. Marquette and C. Quesne while, for k = 0 and n = 3, Φ0,3,0 = Ψ0,3,0, Φ0,3,1 = Ψ0,3,1 − 1 2λ Ψ0,3,0, Φ0,3,2 = Ψ0,3,2 + 3 20λ2 Ψ0,3,0 − 1 2λ Ψ0,3,1, Φ0,3,3 = Ψ0,3,3 − 1 30λ3 Ψ0,3,0 + 3 20λ2 Ψ0,3,1 − 1 2λ Ψ0,3,2, Φ0,3,4 = Ψ0,3,4 − 1 300λ4 Ψ0,3,0 + 1 60λ3 Ψ0,3,1 − 1 20λ2 Ψ0,3,2, Φ0,3,5 = Ψ0,3,5, Φ0,3,6 = Ψ0,3,6. In this section, we have pointed out that the associated functions of the Jordan blocks for nondiagonalizable non-Hermitian Hamiltonians require much more involved calculations when going from two to three dimensions. The use of new ladder operators has, nevertheless, allowed us to conjecture the form of their general normalization coefficient and to enhance the need for an orthogonalization of the basis. 7 Conclusion In the present paper, we have demonstrated that three sets of canonical ladder operators exist for the three-dimensional nonseparable and nondiagonalizable pseudo-Hermitian oscillator of [2]. They can be introduced from their action on the wavefunctions belonging to the lower lattice in Figure 1. These ladder operators have allowed us to show the existence of a nine-dimensional hidden symmetry algebra, which can be written in terms of gl(3) generators. The latter can be expressed in terms of bosonic operators in a nonstandard realization, which serve to embed gl(3) into an sp(6) algebra, as well as into an osp(1/6) superalgebra. Furthermore, we have connected the hidden symmetry algebra with the integrals responsible for the superintegrability of the model and established that the latter generate a cubic algebra. The ladder operators have served to construct the associated functions completing the Jordan blocks, whose dimension has been established. We have also presented the action of these ladder operators on the associated functions, proved that the latter are eigenfunctions of the gl(3) linear Casimir operator, and written a subset of associated functions as multivariate polynomials. Finally, we have studied in detail the construction of an extended biorthogonal basis and shown that its structure is more complicated than that considered in [15] for the corresponding two-dimensional model. Nevertheless, we have been able to conjecture the form of the associated function normalization coefficient and to establish the need for an orthogonalization of the basis. The results obtained here point out some similarities between the present pseudo-Hermitian oscillator and the usual three-dimensional oscillator, but also indicate their very different nature. The method established in this paper may allow a broader understanding of these pseudo- Hermitian models. So far, only particular examples have been considered and no classification has been provided. Our understanding of the underlying hidden symmetry algebra may allow to provide other ideas to obtain and classify such models. Studies of supersymmetric quantum mechanics, superintegrability, separation of variables and hidden algebra have been restricted mostly to models whose states are eigenstates of the Hamiltonian. The present paper points out that models with Jordan blocks have features that make them interesting from a mathematical physics perspective. Among open problems, there is the application of those methods to pseudo- Hermitian anharmonic oscillator models [9]. Ladder Operators and Hidden Algebras for Shape Invariant. II 23 A gl(3) generators in terms of ladder operators The purpose of this appendix is to list the expressions of the gl(3) generators Eij in terms of the ladder operators A±, B±, and C±: E11 = − 1 2λ C+C− + 1 2 , E22 = λ 2g2 B+B− + 1 2λ C+C− + 1 2g (B+C− + C+B−) + 1 2 , E33 = − g2 2λ3 A+A− − λ 2g2 B+B− − 1 2λ C+C− − 1 2λ (A+B− +B+A−) − g 2λ2 (A+C− + C+A−)− 1 2g (B+C− + C+B−) + 1 2 , E12 = i ( 1 2λ C+C− + 1 2g C+B− ) , E21 = i ( 1 2λ C+C− + 1 2g B+C− ) , E13 = − 1 2λ C+C− − 1 2g C+B− − g 2λ2 C+A−, E31 = − 1 2λ C+C− − 1 2g B+C− − g 2λ2 A+C−, E23 = i ( λ 2g2 B+B− + 1 2λ C+C− + 1 2λ B+A− + 1 2g B+C− + g 2λ2 C+A− + 1 2g C+B− ) , E32 = i ( λ 2g2 B+B− + 1 2λ C+C− + 1 2λ A+B− + 1 2g C+B− + g 2λ2 A+C− + 1 2g B+C− ) . B Polynomials fn,p q (u,w) for some low values of p and q The purpose of this appendix is to list some examples of polynomials fn,pq (u,w), introduced in Section 5.2. For p = 1: q = 1, f (n,1) 1 = −4gw, q = 2, f (n,1) 2 = −4g2. For p = 2: q = 1, f (n,2) 1 = −8g2λu, q = 2, f (n,2) 2 = 16g2(w2 − λ), q = 3, f (n,2) 3 = 32g3w, q = 4, f (n,2) 4 = 16g4. For p = 3: q = 2, f (n,3) 2 = 96g3λuw, q = 3, f (n,3) 3 = −32g3 ( −3λgu+ 2w3 − 6λw ) , q = 4, f (n,3) 4 = −192g4 ( w2 − λ ) , q = 5, f (n,3) 5 = −192g5w, q = 6, f (n,3) 6 = −64g6. 24 I. Marquette and C. Quesne Acknowledgments I. Marquette was supported by Australian Research Council Future Fellowhip FT180100099. C. Quesne was supported by the Fonds de la Recherche Scientifique - FNRS under Grant Number 4.45.10.08. References [1] Bagchi B.K., Supersymmetry in quantum and classical mechanics, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 116, Chapman & Hall/CRC, Boca Raton, FL, 2001. [2] Bardavelidze M.S., Cannata F., Ioffe M.V., Nishnianidze D.N., Three-dimensional shape invariant non- separable model with equidistant spectrum, J. Math. Phys. 54 (2013), 012107, 11 pages, arXiv:1212.4805. 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Physics 213 (1992), 74–101. https://doi.org/10.1063/1.4774292 https://arxiv.org/abs/1212.4805 https://doi.org/10.1080/00107500072632 https://arxiv.org/abs/quant-ph/0501052 https://doi.org/10.1088/0034-4885/70/6/R03 https://arxiv.org/abs/hep-th/0703096 https://doi.org/10.1103/PhysRevLett.80.5243 https://doi.org/10.1103/PhysRevLett.80.5243 https://arxiv.org/abs/physics/9712001 https://doi.org/10.1016/S0375-9601(01)00146-3 https://doi.org/10.1016/S0375-9601(01)00146-3 https://arxiv.org/abs/quant-ph/0101095 https://doi.org/10.1088/0305-4470/35/6/305 https://arxiv.org/abs/hep-th/0201080 https://arxiv.org/abs/hep-th/0201080 https://doi.org/10.1063/1.3298675 https://arxiv.org/abs/0910.0590 https://doi.org/10.1088/1751-8113/45/29/295303 https://arxiv.org/abs/1206.4013 https://doi.org/10.1016/0370-1573(94)00080-M https://arxiv.org/abs/hep-th/9405029 https://arxiv.org/abs/hep-th/9607161 https://doi.org/10.1007/s11232-006-0092-7 https://arxiv.org/abs/hep-th/0512110 https://doi.org/10.1007/978-3-642-61194-0 https://doi.org/10.3842/SIGMA.2022.004 https://arxiv.org/abs/2010.15273 https://doi.org/10.1063/1.1514834 https://arxiv.org/abs/math-ph/0207009 https://doi.org/10.1063/1.1418246 https://arxiv.org/abs/math-ph/0107001 https://doi.org/10.1063/1.1540714 https://arxiv.org/abs/math-ph/0301030 https://doi.org/10.1142/S0219887810004816 https://doi.org/10.1142/S0219887810004816 https://arxiv.org/abs/0810.5643 https://doi.org/10.1016/S0375-9601(02)01359-2 https://doi.org/10.1103/RevModPhys.15.175 https://doi.org/10.1016/0003-4916(92)90284-S 1 Introduction 2 Shape invariant model with quadratic complex interaction 3 Construction of additional sets of ladder operators 4 Construction of the hidden symmetry algebra 4.1 Connection with gl(3) and bosonic operators 4.2 Superintegrability and cubic algebra 5 Nondiagonalizability and construction of associated functions 5.1 Algebraic construction of associated functions 5.2 Some associated functions in terms of multivariate polynomials 5.3 Action of the ladder operators and of the gl(3) Casimir operator on Psi_{k,n,m} 6 Construction of an extended biorthogonal basis 6.1 Calculation of the normalization coefficient 6.1.1 Case k=0 and any n 6.1.2 Case n=0 and any k 6.1.3 Case any k and any n 6.2 Orthogonality of the basis 7 Conclusion A gl(3) generators in terms of ladder operators B Polynomials f_q^{n,p}(u,w) for some low values of p and q References
id	nasplib_isofts_kiev_ua-123456789-211540
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-17T09:27:48Z
publishDate	2022
publisher	Інститут математики НАН України
record_format	dspace
spelling	Marquette, Ian Quesne, Christiane 2026-01-05T12:30:23Z 2022 Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model. Ian Marquette and Christiane Quesne. SIGMA 18 (2022), 005, 24 pages 1815-0659 2020 Mathematics Subject Classification: 81Q05; 81Q60; 81R12; 81R15 arXiv:2010.15276 https://nasplib.isofts.kiev.ua/handle/123456789/211540 https://doi.org/10.3842/SIGMA.2022.005 A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remain to be studied. We present a set of six operators { ⁺⁻, ⁺⁻, ⁺⁻} that can be combined to build a (3) hidden algebra. The latter can be embedded in an (6) algebra, as well as in an (1/6) superalgebra. The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state. We present the action of these operators and study the construction of an extended biorthogonal basis. These rely on establishing various nontrivial polynomial and commutator identities. We also make a connection between the hidden symmetry and the underlying superintegrability property of the model. Interestingly, the integrals generate a cubic algebra. This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences. I. Marquette was supported by the Australian Research Council Future Fellowship FT180100099. C. Quesne was supported by the Fonds de la Recherche Scientifique- FNRS under Grant Number 4.45.10.08. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model Article published earlier
spellingShingle	Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model Marquette, Ian Quesne, Christiane
title	Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model
title_full	Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model
title_fullStr	Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model
title_full_unstemmed	Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model
title_short	Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model
title_sort	ladder operators and hidden algebras for shape invariant nonseparable and nondiagonalizable modelswith quadratic complex interaction. ii. three-dimensional model
url	https://nasplib.isofts.kiev.ua/handle/123456789/211540
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Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model

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